Advertisement

aequationes mathematicae

, Volume 39, Issue 2–3, pp 149–160 | Cite as

Trivialization of fans in planar ternary rings with rational prime field

  • Franz B. Kalhoff
Research Papers
  • 25 Downloads

Summary

Making use of the intimate relations between real places and orderings, we continue studying the spaces of orderings of planar ternary rings (PTRs) with rational prime field. As in the classical case, given a preorderingS of such a PTRT, then the productA S of all natural place ringsA p associated to the orderingsP ∈ X/S is itself a place ring ofT. In particular, ifS is a nontrivial fan ofT thenA s ≠ T. Thus L. Bröcker's celebrated theorem on the trivialization of fans also applies to our setting:

For any fan Sof a PTRT with rational prime field there exists a place µ: T → T′ ∪ {∞} such that the push down S′:= μ(S)\{0,∞} of S is a trivial fan of T′.

Further, Bröcker's global stability formula and, by means of an approximation theorem, some classical, valuation theoretic characterizations of SAP-preorderings and fans also extend to PTRs with rational prime field.

AMS (1980) subject classification

Primary 51G05 Secondary 11E81, 12K05, 12K10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    André, J.,Über Homomorphismen projektiver Ebenen. Abh. Math. Sem. Univ. Hamburg34 (1969/70), 98–114.Google Scholar
  2. [2]
    Bröcker, L.,Zur Theorie der quadratischen Formen über formal-reellen Körpern. Math. Ann.210 (1974), 233–256.CrossRefGoogle Scholar
  3. [3]
    Bröcker, L.,Characterization of Fans and Hereditarily Pythagorean Fields. Math. Z.151 (1976), 149–163.CrossRefGoogle Scholar
  4. [4]
    Brown, R.,Superpythagorean Fields. J. Algebra42 (1976), 483–494.CrossRefGoogle Scholar
  5. [5]
    Bruck, R. H.,A Survey of Binary Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete20, Springer-Verlag, Berlin (1966), 2nd Edn. (1st Edn. 1958).Google Scholar
  6. [6]
    Crampe, S.,Angeordnete projektive Ebenen. Math. Z.69 (1958), 435–462.CrossRefGoogle Scholar
  7. [7]
    Hartmann, P.,Topologisierung projektiver Ebenen durch Epimorphismen. Dissertation. LMU München, 1986.Google Scholar
  8. [8]
    Hartmann, P.,Die Stellentopologie projektiver Ebenen und Lenz-topologische Ebenen. Geom. Dedicata26 (1988), 259–272.CrossRefGoogle Scholar
  9. [9]
    Hartmann, P. andPrieß-Crampe, S.,Zur Existenz und Fortsetzbarkeit von o-Epimorphismen projektiver Ebenen. Abh. Math. Sem. Univ. Hamburg58 (1988), 149–168.Google Scholar
  10. [10]
    Hartmann, P. andPrieß-Crampe, S.,Zur Fortsetzbarkeit von o-Stellen angeordneter Divisions algebren. Arch Math.51 (1988), 178–180.CrossRefGoogle Scholar
  11. [11]
    Kalhoff, F.,Eine Kennzeichnung anordnungsfähiger Ternärkörper. J. Geometry31 (1988), 100–113.CrossRefGoogle Scholar
  12. [12]
    Kalhoff, F.,Über Präordnungen von Ternärkörpern. Abh. Math. Sem. Univ. Hamburg58 (1988), 5–13.Google Scholar
  13. [13]
    Kalhoff, F.,Spaces of orderings and Witt rings of planar ternary rings. J. Pure Appl. Algebra58 (1989), 169–180.CrossRefGoogle Scholar
  14. [14]
    Kalhoff, F.,Über Stellen und Präordnungen von Ternärkörpern. Geom. Dedicata27 (1988), 137–151.CrossRefGoogle Scholar
  15. [15]
    Kalhoff, F.,Uniform valuations on Planar Ternary Rings. Geom. Dedicata28 (1988), 337–348.CrossRefGoogle Scholar
  16. [16]
    Kalhoff, F.,On order compatible places or near fields. Resultate Math.15 (1989), 66–74.Google Scholar
  17. [17]
    Kalhoff, F.,Some local-global principles for ordered projective planes. Geom. Dedicata32 (1989), 59–79.CrossRefGoogle Scholar
  18. [18]
    Kalhoff, F.,The holomorphy rings of planar ternary rings with rational prime field, to appear.Google Scholar
  19. [19]
    Karzel, H.,Ordnungsfunktionen in nicht desarguesschen projektiven Geometrien. Math. Z.62 (1955), 268–291.CrossRefGoogle Scholar
  20. [20]
    Karzel, H.,Anordnungsfragen in Ternären Ringen und allgemeinen projektiven und affine Ebenen. Algebraical and topological Foundations of Geometry, Oxford, 1962, 71–86.Google Scholar
  21. [21]
    Lam, T. Y.,Orderings, valuations and quadratic forms. CBMS Regional Conference Series in Mathematics52 (1983), American Mathematical Society, Providence.Google Scholar
  22. [22]
    Marshall, M. A.,Classification of finite spaces of orderings. Canad. J. Math.31 (1979), 320–330.Google Scholar
  23. [23]
    Mathiak, K.,Valuations of Skew Fields and Projective Hjelmselev Spaces. Lecture Notes in Math.1175, Springer, Berlin, 1980.Google Scholar
  24. [24]
    Pickert, G.,Projektive Ebenen. Springer, Berlin—Heidelberg—New York, 2nd Edn. 1975 (1st Edn. 1955).Google Scholar
  25. [25]
    Prieß-Crampe, S.,Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen. Springer, Berlin—Heidelberg—New York, 1983.Google Scholar
  26. [26]
    Prieß-Crampe, S.,o-Epimorphismen projektiver Ebenen. Geom. Dedicata22 (1987), 21–37.Google Scholar
  27. [27]
    Skornjakov, L. A.,Homomorphisms of projective planes and T-homomorphisms of ternaries (Russian). Math. Sb.43 (1957), 285–294.Google Scholar
  28. [28]
    Tschimmel, A.,Über Anordnungsräume von Schiefkörpern. Dissertation Münster 1981.Google Scholar
  29. [29]
    Tschimmel, A.,Lokal-Global Prinzipien für Anordnungen bewerteter Schiefkörper. Arch. Math.44 (1985), 48–58.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • Franz B. Kalhoff
    • 1
  1. 1.Fachbereich MathematikUniversität DortmundDortmund 50West Germany

Personalised recommendations